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Properties

Label 2-2900-2900.1887-c0-0-0
Degree $2$
Conductor $2900$
Sign $-0.304 + 0.952i$
Analytic cond. $1.44728$
Root an. cond. $1.20303$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0448 − 0.998i)2-s + (−0.995 + 0.0896i)4-s + (−0.722 − 0.691i)5-s + (0.134 + 0.990i)8-s + (−0.473 + 0.880i)9-s + (−0.657 + 0.753i)10-s + (1.05 + 0.729i)13-s + (0.983 − 0.178i)16-s + (0.617 − 1.90i)17-s + (0.900 + 0.433i)18-s + (0.781 + 0.623i)20-s + (0.0448 + 0.998i)25-s + (0.681 − 1.08i)26-s + (0.880 − 0.473i)29-s + (−0.222 − 0.974i)32-s + ⋯
L(s)  = 1  + (−0.0448 − 0.998i)2-s + (−0.995 + 0.0896i)4-s + (−0.722 − 0.691i)5-s + (0.134 + 0.990i)8-s + (−0.473 + 0.880i)9-s + (−0.657 + 0.753i)10-s + (1.05 + 0.729i)13-s + (0.983 − 0.178i)16-s + (0.617 − 1.90i)17-s + (0.900 + 0.433i)18-s + (0.781 + 0.623i)20-s + (0.0448 + 0.998i)25-s + (0.681 − 1.08i)26-s + (0.880 − 0.473i)29-s + (−0.222 − 0.974i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $-0.304 + 0.952i$
Analytic conductor: \(1.44728\)
Root analytic conductor: \(1.20303\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2900} (1887, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2900,\ (\ :0),\ -0.304 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9099351095\)
\(L(\frac12)\) \(\approx\) \(0.9099351095\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0448 + 0.998i)T \)
5 \( 1 + (0.722 + 0.691i)T \)
29 \( 1 + (-0.880 + 0.473i)T \)
good3 \( 1 + (0.473 - 0.880i)T^{2} \)
7 \( 1 + (0.433 + 0.900i)T^{2} \)
11 \( 1 + (-0.834 - 0.550i)T^{2} \)
13 \( 1 + (-1.05 - 0.729i)T + (0.351 + 0.936i)T^{2} \)
17 \( 1 + (-0.617 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.880 - 0.473i)T^{2} \)
23 \( 1 + (0.512 + 0.858i)T^{2} \)
31 \( 1 + (0.657 + 0.753i)T^{2} \)
37 \( 1 + (-0.692 - 0.372i)T + (0.550 + 0.834i)T^{2} \)
41 \( 1 + (0.482 + 0.0764i)T + (0.951 + 0.309i)T^{2} \)
43 \( 1 + (-0.222 - 0.974i)T^{2} \)
47 \( 1 + (-0.963 - 0.266i)T^{2} \)
53 \( 1 + (0.365 + 1.78i)T + (-0.919 + 0.393i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.307 + 0.767i)T + (-0.722 + 0.691i)T^{2} \)
67 \( 1 + (-0.266 - 0.963i)T^{2} \)
71 \( 1 + (-0.963 - 0.266i)T^{2} \)
73 \( 1 + (-1.80 + 0.498i)T + (0.858 - 0.512i)T^{2} \)
79 \( 1 + (0.998 + 0.0448i)T^{2} \)
83 \( 1 + (-0.880 + 0.473i)T^{2} \)
89 \( 1 + (-1.10 + 1.21i)T + (-0.0896 - 0.995i)T^{2} \)
97 \( 1 + (-0.117 - 1.31i)T + (-0.983 + 0.178i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.775494638587800190431396683490, −8.161893913749298587684944199599, −7.57126155391865378091865009163, −6.38998175259044712500082027514, −5.07740995113795537181734326898, −4.92828023221201249111234477361, −3.83527683436592419614482281819, −3.06677115606541664008932402170, −1.98714094078872027764163021696, −0.75659580845000623523641325349, 1.09131827875647199533482462828, 3.09840559840042552469264234654, 3.67560380570495979437207211708, 4.42350735265841333566575216018, 5.76293255988031741428222291642, 6.12386848959627978572256404017, 6.81351399133528659561384676686, 7.79023752365784986286131705708, 8.264836841905075099206080422452, 8.831191046670444425617909043442

Graph of the $Z$-function along the critical line